The technique for this sort of the multiplier circuit has so far been proposed by the present inventor. As the mathematical basis for obtaining a product of two signals, a quarter-square technique is known. The generalized equation has been presented by the present inventor as the following equations (1) and (2):(ax+by)2+{(a−c)x+(b−1/c)y}2−{(a−c)x+by}2−{ax+(b−1/c)y}2=2xy  (1)(ax+by+z)2+{(a−c)x+(b−1/c)y+z}2−{(a−c)x+by+z}2−{ax+(b−1/c)y+z}2=2xy  (2)
where a, b and c are constants and x, y and z are variables.
It suffices here to set the constants a, b and c, with x being a first input signal, y being a second input signal and z being an arbitrary variable.
For example, in a multiplier circuit, proposed by the presents inventor, a=b=c=1 is used, such that(x+y)2−y2−x2=2xy  (3)or(x+y+z)2+z2−(y+z)2−(x+z)2=2xy  (4)
In a multiplier circuit, proposed by Bult et al., a=b=1/2 and c=1 are used, such that(x/2+y/2)2+(−x/2−y/2)2−(−x/2+y/2)2−(x/2−y/2)2=2xy  (5)or(x/2+y/2+z)2+(−x/2−y/2+z)2−(−x/2+y/2+z)2−(x/2−y/2+z)2=2xy  (6)
In another multiplier circuit, proposed by Bult, a=1/2 and b=c=1 are used, such that(x/2+y)2+(−x/2)2−(−x/2+y)2−(x/2)2=2xy  (7)or(x/2+y+z)2+(−x/2+z)2−(−x/2+y+z)2−(x/2+z)2=2xy  (8)
While the equation (3) is noticed from time to time, the following equation (9):(x+y)2−(x−y)2=4xy  (9)obtained on re-arranging the equation (5) for simplification is generally called the quarter-square technique.
As indicated in the foregoing, a multiplier circuit is implemented by combining a summation circuit (subtraction circuit) and a multiplier core circuit having the function of a squaring circuit.
FIG. 1 is a diagram showing a configuration of a multiplier core circuit formed by a quadritail cell. The multiplier core circuit includes MOS transistors M1, M2, M3 and M4, having sources coupled together and connected to a current source. The drains of the transistors M1 and M2 are coupled together, while the drains of the transistors M3 and M4 are also coupled together.
Assuming that the voltage V1, applied to the gate of the MOS transistor M1, is expressed as VCM+aVx+bVy,
the voltage V2 applied to the gate of the MOS transistor M2, is expressed as VCM+(a−c)Vx+(b−1/c)Vy,
the voltage V3, applied to the gate of the MOS transistor M3, is expressed as VCM+(a−c)Vx+bVy and
the voltage V4, applied to the gate of the MOS transistor M4, is expressed as VCM+aVx+(b−1/c)Vy,
drain currents ID1, ID2, ID3 and ID4 of the MOS transistors M1, M2, M3 and M4 may respectively be expressed by the following equations (10) to (13):ID1=β(VCM+aVx+bVy−VSQ−VTH)2  (10)ID2=β{VCM+(a−c)Vx+(b−1/c)Vy−VSQ−VTH}2  (11)ID3=β{VCM+(a−c)Vx+bVy−VSQ−VTH}2  (12)ID4=β{VCM+aVx+(b−1/c)Vy−VSQ−VTH}2  (13)
In the above equations, β is a transcondunctance parameter of a unit transistor and is expressed asβ=(1/2)μ(W/L)(∈x/tox)  (14)where μ denotes an effective electron mobility, ∈x denotes a dielectric constant of a gate insulating film, tox denotes a film thickness of a gate insulating film, W denotes channel width and L denotes a channel length.
If we put z=VCM−VSQ−VTH, a differential output current ΔI of the multiplier core circuit may be found asΔI=(ID1+ID2)−(ID3+ID4)=2βVx×Vy  (15)from which it is seen that the quadritail cell represents a multiplier core circuit.
It is noted that, from the following condition of the tail current:ID1+ID2+ID3+ID4=I0  (16)and from the input voltage at which transistors of the quadritail cell are pinched-off, the differential output current ΔI may be found in accordance with the following equation (17):
                                                                        Δ                ⁢                                                                  ⁢                I                            =                            ⁢                                                (                                                            I                                              D                        ⁢                                                                                                  ⁢                        1                                                              +                                          I                                              D                        ⁢                                                                                                  ⁢                        2                                                                              )                                -                                  (                                                            I                                              D                        ⁢                                                                                                  ⁢                        3                                                              +                                          I                                              D                        ⁢                                                                                                  ⁢                        4                                                                              )                                                                                                        =                            ⁢                              {                                                                                                                            ⁢                                                  2                          ⁢                          β                          ⁢                                                                                                          ⁢                                                      V                            x                                                    ⁢                                                                                    V                              y                                                        ⁡                                                          (                                                                                                                                    V                                    x                                    2                                                                    +                                                                      V                                    y                                    2                                                                    +                                                                                                                                                                                        V                                        x                                                                            ⁢                                                                              V                                        y                                                                                                                                                                                                                ≤                                                                                                      I                                    0                                                                                                        2                                    ⁢                                    β                                                                                                                              )                                                                                                                                                                                                                                                      ⁢                                                                                                            4                              3                                                        ⁢                            β                            ⁢                                                                                                                  ⁢                                                          V                              x                                                        ⁢                                                          V                              y                                                                                -                                                                                    1                              9                                                        ⁢                                                          sgn                              (                                                                                                V                                  x                                                                ⁢                                                                  V                                  y                                                                                            )                                                        ⁢                                                          {                                                                                                3                                  ⁢                                                                      I                                    0                                                                                                  +                                                                  2                                  ⁢                                  β                                  ⁢                                                                                                                                          ⁢                                                                                                            (                                                                                                                                                                                                              V                                            x                                                                                                                                                                    +                                                                                                                                                                        V                                            y                                                                                                                                                                                                        )                                                                        2                                                                                                  -                                                                                                                                                                                                                                                                                                                ⁢                                                      4                            ⁢                            β                            ⁢                                                                                                                  ⁢                                                          (                                                                                                                                                                      V                                    x                                                                                                                                    +                                                                                                                                        V                                    y                                                                                                                                                                )                                                                                }                                                ⁢                                                                                                                                            3                                ⁢                                                                  I                                  0                                                                                            β                                                        -                                                          2                              ⁢                                                                                                (                                                                                                                                                                                          V                                        x                                                                                                                                                    +                                                                                                                                                        V                                        y                                                                                                                                                                                    )                                                                2                                                                                      +                                                          6                              ⁢                                                                                                                                                                    V                                    x                                                                    ⁢                                                                      V                                    y                                                                                                                                                                                                                                                                                                                                                                                      ⁢                                                  (                                                                                    V                              x                              2                                                        +                                                          V                              y                              2                                                        +                                                                                                                                                                                              V                                    x                                                                    ⁢                                                                      V                                    y                                                                                                  |                                                                  ≥                                                                                                            I                                      0                                                                                                              2                                      ⁢                                      β                                                                                                        ≥                                                                                                            V                                      x                                      2                                                                        +                                                                                                                  V                                        y                                        2                                                                            2                                                                        -                                                                                                                                                                                                  V                                          x                                                                                ⁢                                                                                  V                                          y                                                                                                                                                                                                                                                                                      )                                                                                                                                                                                                                                                      ⁢                                                                              (                                                          β                              ⁢                                                                                                                          ⁢                                                              V                                y                                                            ⁢                                                                                                                                                                          2                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    y                                    2                                                                                                                                                        )                                                    ⁢                                                      (                                                                                                                            V                                  x                                  2                                                                +                                                                                                      V                                    y                                    2                                                                    2                                                                -                                                                                                                                                                              V                                      x                                                                        ⁢                                                                          V                                      y                                                                                                                                                                                                    ≥                                                                                                I                                  0                                                                                                  2                                  ⁢                                  β                                                                                                                      )                                                                                                                                                                                                      (        17        )            
It should be noted that here the errors in the Publications of the related art, inclusive of Publication 1, have been corrected in the above statements.
Thus, as for an input voltage at which a MOS transistor operating in the saturation region is not pinched-off, the product Vx×Vy, where Vx, Vy denote two input voltages, is obtained, as indicated by the equation (17), with the quadritail cell operating as a multiplier core circuit.
As the input voltage becomes higher, transistors that make up the quadritail cell get pinched off, and the multiplication characteristic of the circuit becomes deviated from the ideal characteristic.
FIG. 2 shows differential output current characteristics of the quadritail cell as calculated with the equation (17) with Vy as a parameter. In FIG. 2, the input voltage range for the regular operation, shown by the equation (17), is indicated by a broken line.
Since the quadritail cell has only one tail current, the multiplier circuit suffers from limiting with increase in the input signal.
The operating range of the quadritail cell is shown in FIG. 3, in which errors of the related art Publication have been corrected. The operating range has two heart shapes inverted relative to and partially overlapped with each other, with the input voltage range for the regular operation (Vx,Vy) being of a rounded lozenge corresponding to the overlapped region which is shown shaded in FIG. 3.
A voltage summation circuit is now described. FIG. 4 shows a configuration of a voltage summation circuit implemented using two MOS differential pairs (M1, M2) and (M5, M6). The sources of the n-channel MOS differential pair (M1, M2) are coupled together and connected to a constant current source, and VIN is differentially applied to their gates. The sources of the n-channel MOS differential pair (M5, M6) are coupled together and connected to another current source. The drains of the n-channel MOS differential pair (M3, M4) are respectively connected to the drains of the MOS transistors M1 and M6. The drain of the MOS transistor M2 is connected to a power supply VDD, whilst the drain and the gate of the MOS transistor M5 are respectively connected to the power supply VDD and to VREF. The drain and the gate of the MOS transistor M6 are coupled together. The drain voltage (=gate voltage) of the MOS transistor M6 is to be VOUT. The drain current ID1 of the MOS transistor M1 is equal to the drain current ID6 of the MOS transistor M6 (ID1=ID6), and the gate-to-source voltage VGS1 of the transistor M1 is equal to the gate-to-source voltage VGS6 of the transistor M6 (VGS1=VGS6). Since ID1+ID2=ID5+ID6=IOO, ID2=ID5. The gate-to-source voltage VGS2 of the transistor M2 is equal to the gate-to-source voltage VGS5 of the transistor M5 (VGS2=VGS5).VOUT−VREF=VGS6−VGS5=VGS1−VGS2=VIN 
is valid, and hence the output voltage VOUT is expressed by the following equation:VOUT=VREF+VIN  (18)
A multiplier circuit, employing the voltage summation circuit of FIG. 4 as an input circuit, is shown in FIG. 5 (see FIG. 6 of Patent Document 1). It should be noted that, with this active voltage summation circuit, the high frequency operation is restricted by the cut-off frequency fT of the p-channel transistor. It is thus necessary to take the frequency characteristic into consideration. It should also be noted that, with the multiplier circuit, proposed by Bult, a=1/2 and b=c=1. The in-depth analysis in case of integration of the circuit in question may be found in a Wang's thesis. Briefly, there is proposed a multiplier circuit making use of three MOS differential pairs as an input circuit. These three MOS differential pairs perform the role of two voltage summation circuits shown in FIG. 4.
It may be surmised that the circuit is possibly not meritorious because it makes use of a number of MOS differential pairs greater by one than in the case of the circuit proposed by the present inventor (FIG. 5).
[Patent Document 1] Japanese Patent No. 2671872
[Non-Patent Document 1] K. Kimura “An MOS Four-Quadrant Analog Multiplier Based on the Multitail Technique Using a Quadritail Cell as a Multiplier Core”, IEEE Transactions on Circuits and Systems-I, Vol. 42, No. 8, pp. 448-454, August 1995